A133028 -id:A133028 - cp's OEIS frontend

A020492 Balanced numbers: numbers k such that phi(k) (A000010) divides sigma(k) (A000203).

1, 2, 3, 6, 12, 14, 15, 30, 35, 42, 56, 70, 78, 105, 140, 168, 190, 210, 248, 264, 270, 357, 418, 420, 570, 594, 616, 630, 714, 744, 812, 840, 910, 1045, 1240, 1254, 1485, 1672, 1848, 2090, 2214, 2376, 2436, 2580, 2730, 2970, 3080, 3135, 3339, 3596, 3720, 3828

Offset: 1

David W. Wilson

nonn

The quotient A020492(n)/A002088(n) = SummatorySigma/SummatoryTotient as n increases seems to approach Pi^4/36 or zeta(2)^2 [~2.705808084277845]. - Labos Elemer, Sep 20 2004, corrected by Charles R Greathouse IV, Jun 20 2012
If 2^p-1 is prime (a Mersenne prime) then m = 2^(p-2)*(2^p-1) is in the sequence because when p = 2 we get m = 3 and phi(3) divides sigma(3) and for p > 2, phi(m) = 2^(p-2)*(2^(p-1)-1); sigma(m) = (2^(p-1)-1)*2^p hence sigma(m)/phi(m) = 4 is an integer. So for each n, A133028(n) = 2^(A000043(n)-2)*(2^A000043(n)-1) is in the sequence. - Farideh Firoozbakht, Nov 28 2005
Phi and sigma are both multiplicative functions and for this reason if m and n are coprime and included in this sequence then m*n is also in this sequence. - Enrique Pérez Herrero, Sep 05 2010
The quotients sigma(n)/phi(n) are in A023897. - Bernard Schott, Jun 06 2017
There are 544768 balanced numbers < 10^14. - Jud McCranie, Sep 10 2017
a(975807) = 419998185095132. - Jud McCranie, Nov 28 2017

			sigma(35) = 1+5+7+35 = 48, phi(35) = 24, hence 35 is a term.

D. Chiang, "N's for which phi(N) divides sigma(N)", Mathematical Buds, Chap. VI pp. 53-70 Vol. 3 Ed. H. D. Ruderman, Mu Alpha Theta 1984.

Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Jud McCranie, 670314 balanced numbers (first 1000 from T. D. Noe, first 10000 from Donovan Johnson)

Cf. A000010, A000043, A000203, A000668, A011257, A023897, A133028, A291565, A291566, A292422, A351114 (characteristic function).

Positions of 0's in A063514.

[ n: n in [1..3900] | SumOfDivisors(n) mod EulerPhi(n) eq 0 ]; // Klaus Brockhaus, Nov 09 2008

Select[ Range[ 4000 ], IntegerQ[ DivisorSigma[ 1, # ]/EulerPhi[ # ] ]& ]
(* Second program: *)
Select[Range@ 4000, Divisible[DivisorSigma[1, #], EulerPhi@ #] &] (* Michael De Vlieger, Nov 28 2017 *)

select(n->sigma(n)%eulerphi(n)==0,vector(10^4,i,i)) \\ Charles R Greathouse IV, Jun 20 2012

from sympy import totient, divisor_sigma
print([n for n in range(1, 4001) if divisor_sigma(n)%totient(n)==0]) # Indranil Ghosh, Jul 06 2017

from math import prod
from itertools import count, islice
from sympy import factorint
def A020492_gen(startvalue=1): # generator of terms >= startvalue
    for m in count(max(startvalue,1)):
        f = factorint(m)
        if not prod(p**(e+2)-p for p,e in f.items())%(m*prod((p-1)**2 for p in f)):
            yield m
A020492_list = list(islice(A020492_gen(),20)) # Chai Wah Wu, Aug 12 2024

More terms from Farideh Firoozbakht, Nov 28 2005

A134708 Even superperfect numbers divided by 2.

Original entry on oeis.org

1, 2, 8, 32, 2048, 32768, 131072, 536870912, 576460752303423488, 154742504910672534362390528, 40564819207303340847894502572032, 42535295865117307932921825928971026432

Offset: 1

Omar E. Pol, Nov 07 2007, Apr 23 2008

nonn

a(13) and a(14) have 157 and 183 digits respectively. - R. J. Mathar, Jan 07 2008
Largest proper divisor of n-th even superperfect number A061652(n). Also, largest proper divisor of n-th superperfect number A019279(n), if there are no odd superperfect numbers.
Indices of even hexagonal numbers (A014635) that are also even perfect numbers. - Omar E. Pol, Jan 11 2009

			a(5) = 2048 because the 5th even superperfect number is 4096 and 4096/2 = 2048.

Amiram Eldar, Table of n, a(n) for n = 1..18
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos [From _Omar E. Pol_, Jan 11 2009]

Cf. A019279, A061652, A133028.
Cf. A000043.
Cf. A032742, A138882, A139248.
Cf. A000217, A000396, A014635. - Omar E. Pol, Jan 11 2009

A000043 := proc(n) op(n,[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213]) ; end: A061652 := proc(n) 2^(A000043(n)-1) ; end: A134708 := proc(n) A061652(n)/2 ; end: seq(A134708(n),n=1..14) ; # R. J. Mathar, Jan 07 2008

With[{max = 12}, 2^(MersennePrimeExponent[Range[max]] - 2)] (* Amiram Eldar, Oct 21 2024 *)

a(n) = A061652(n)/2.
a(n) = 2^(A000043(n)-2). - Omar E. Pol, Mar 01 2008
a(n) = A032742(A061652(n)). Also, a(n) = A032742(A019279(n)), if there are no odd superperfect numbers.
a(n) = Sum_{x=1..n-th superperfect number} x*(-1)^x. - Juri-Stepan Gerasimov, Jul 21 2009

More terms from R. J. Mathar, Jan 07 2008

A293391 Integers n such that sigma(n)/phi(n) is a perfect square.

Original entry on oeis.org

1, 14, 30, 105, 248, 264, 418, 714, 1485, 3080, 3135, 3596, 3828, 3956, 4064, 5396, 6678, 8636, 10098, 12648, 20026, 20790, 21318, 22152, 23374, 24882, 25714, 26040, 35074, 35343, 39105, 41656, 43890, 44660, 49938, 55154, 56134, 56536, 61344, 71145, 74613, 86304, 87087, 94944

Offset: 1

Jose Arnaldo Bebita Dris, Oct 08 2017

nonn

From Robert Israel, Dec 12 2017: (Start)
Intersection of A011257 and A020492.
If x and y are coprime members of the sequence, then x*y is in the sequence.
Contains all members of A133028 except 3. (End)

			sigma(14)=3*8=24, phi(14)=14*(1/2)*(6/7)=6, sigma(14)/phi(14)=2^2, so 14 is in the list.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 349 terms from Robert Israel)
J. A. B. Dris and C. Leibovici, Why is this sequence not in the OEIS?, October 8 2017.
ProofWiki, Integers whose ratio between sigma and phi is square, misses the second term, 14 as of Dec 2017.

Cf. A000010, A000203, A011257, A020492, A133028, A289336, A289412, A292422.

for n from 1 to 100000 do
    r := numtheory[sigma](n)/numtheory[phi](n) ;
    if issqr(r) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 07 2017

Select[Range[10^5], IntegerQ@ Sqrt[DivisorSigma[1, #]/EulerPhi[#]] &] (* Michael De Vlieger, Dec 08 2017 *)

isok(n) = my(q=sigma(n)/eulerphi(n)); issquare(q) && (denominator(q) == 1); \\ Michel Marcus, Dec 07 2017; corrected Sep 21 2019

a(n) = sigma(n)/phi(n) = m^2, for some integer m.

A139045 Largest proper divisor of the Fibonacci numbers > 1.

Original entry on oeis.org

1, 1, 1, 4, 1, 7, 17, 11, 1, 72, 1, 29, 305, 329, 1, 1292, 113, 2255, 5473, 199, 1, 23184, 15005, 521, 98209, 105937, 1, 416020, 2417, 726103, 1762289, 3571, 1845493, 7465176, 330929, 1056437, 31622993, 34111385, 59369, 133957148, 1, 233802911, 567451585

Offset: 3

Omar E. Pol, Apr 23 2008

nonn

See the list of divisors of positive Fibonacci numbers in the triangle A133021.
See the largest proper divisor of n in A032742.
Fibonacci(1)=Fibonacci(2)=1 do not have proper divisors. - Emeric Deutsch, May 18 2008

			a(9) = 17 because the 9th Fibonacci number is 34 and the divisors of 34 are 1, 2, 17, 34, then the largest proper divisor of 34 is 17.

Amiram Eldar, Table of n, a(n) for n = 3..1000

Cf. A000045, A032742, A060383, A133021, A133028, A134708.

with(combinat): with(numtheory): a:=proc(n) options operator, arrow: op(tau(fibonacci(n))-1, divisors(fibonacci(n))) end proc: seq(a(n),n=3..40); # Emeric Deutsch, May 18 2008
# second Maple program:
a:= n-> (f-> f/min(numtheory[factorset](f)))((<<0|1>, <1|1>>^n)[1, 2]):
seq(a(n), n=3..47);  # Alois P. Heinz, Sep 03 2019

lpd[n_]:=Divisors[n][[-2]]; lpd/@(Fibonacci[Range[3,40]]) (* Harvey P. Dale, Mar 29 2015 *)

a(n) = A032742(A000045(n)).

a(n) = A000045(n)/A060383(n). - Alois P. Heinz, Sep 03 2019

More terms from Emeric Deutsch, May 18 2008

A134705 a(n) = n-th perfect number divided by 2^n.

Original entry on oeis.org

3, 7, 62, 508, 1048448, 134216704, 1073739776, 9007199250546688, 5192296858534827626278696515534848, 187072209578355573530071658285452771612302071824384

Offset: 1

Omar E. Pol, Nov 07 2007

nonn

			a(3)=62 because the 3rd perfect number is 496 and 2^3=8 and 496/8=62.

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000396, A133028.

 perfp := [1,2,4,6,12,16,18,30,60,88,106,126,520,606,1278,2202,2280,3216,4252] : A000396 := proc(n) global perfp ; 2^op(n,perfp)*(2^(op(n,perfp)+1)-1) ; end: A134705 := proc(n) A000396(n)/2^n ; end: seq(A134705(n),n=1..12) ; # R. J. Mathar

a(n)=A000396(n)/(2^n).

More terms from R. J. Mathar, Jan 07 2008

A134709 Even superperfect numbers divided by 2, minus 1.

Original entry on oeis.org

0, 1, 7, 31, 2047, 32767, 131071, 536870911, 576460752303423487, 154742504910672534362390527, 40564819207303340847894502572031, 42535295865117307932921825928971026431

Offset: 1

Omar E. Pol, Nov 07 2007

nonn

			a(5)=2047 because the 5th even superperfect number is 4096 and (4096/2)-1=2047.

Cf. A019279, A061652, A133028.

a(n) = (A061652(n)/2) - 1.

More terms from Jinyuan Wang, Mar 14 2020

A134710 a(n) = n-th even superperfect number divided by 2^n.

Original entry on oeis.org

1, 1, 2, 4, 128, 1024, 2048, 4194304, 2251799813685248, 302231454903657293676544, 39614081257132168796771975168, 20769187434139310514121985316880384

Offset: 1

Omar E. Pol, Nov 07 2007

nonn

a(13) and a(14) have 153 and 179 digits respectively and are too large to include here. - R. J. Mathar, Jan 07 2008

			a(5) = 128 because the 5th even superperfect number is 4096 and 2^5 = 32 and 4096/32 = 128.

Amiram Eldar, Table of n, a(n) for n = 1..18
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000043, A000396, A000668, A019279, A061652 (even superperfect numbers), A133028.

A000043 := proc(n) op(n,[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213]) ; end: A061652 := proc(n) 2^(A000043(n)-1) ; end: A134710 := proc(n) A061652(n)/2^n ; end: seq(A134710(n),n=1..14) ; # R. J. Mathar, Jan 07 2008

With[{max = 12}, 2^(MersennePrimeExponent[Range[max]] - Range[max] - 1)] (* Amiram Eldar, Oct 21 2024 *)

a(n) = A061652(n)/(2^n).

a(n) = 2^(A000043(n)-n-1). - Amiram Eldar, Oct 21 2024

More terms from R. J. Mathar, Jan 07 2008

A134712 Base-2 logarithm of (n-th even superperfect number divided by 2^n).

Original entry on oeis.org

0, 0, 1, 2, 7, 10, 11, 22, 51, 78, 95, 114, 507, 592, 1263, 2186, 2263, 3198, 4233, 4402, 9667, 9918, 11189, 19912, 21675, 23182, 44469, 86214, 110473, 132018, 216059, 756806, 859399, 1257752, 1398233, 2976184, 3021339, 6972554, 13466877

Offset: 1

Omar E. Pol, Nov 07 2007

nonn

			a(5) = 7 because the 5th even superperfect number is 4096, 2^5 = 32, 4096/32 = 128 and log_2(128) = 7 (because 2^7 = 128).

Amiram Eldar, Table of n, a(n) for n = 1..48
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000043, A000396, A000668, A019279, A061652, A090748, A133028.

With[{max = 48}, MersennePrimeExponent[Range[max]] - Range[max] - 1] (* Amiram Eldar, Oct 21 2024 *)

a(n) = log_2(A061652(n)/(2^n)) = A000043(n) - n - 1 = A090748(n) - n.

A134713 Base-2 logarithm of (n-th even superperfect number divided by 2^n), plus 1.

Original entry on oeis.org

1, 1, 2, 3, 8, 11, 12, 23, 52, 79, 96, 115, 508, 593, 1264, 2187, 2264, 3199, 4234, 4403, 9668, 9919, 11190, 19913, 21676, 23183, 44470, 86215, 110474, 132019, 216060, 756807, 859400, 1257753, 1398234, 2976185, 3021340, 6972555, 13466878

Offset: 1

Omar E. Pol, Nov 07 2007

nonn

			a(5) = 8 because the 5th even superperfect number is 4096, 2^5 = 32, 4096/32 = 128, log_2(128) = 7 (because 2^7 = 128) and 7+1 = 8.

Amiram Eldar, Table of n, a(n) for n = 1..48
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000043, A000396, A000668, A019279, A061652, A090748, A133028.

With[{max = 48}, MersennePrimeExponent[Range[max]] - Range[max]] (* Amiram Eldar, Oct 21 2024 *)

a(n) = 1 + log_2(A061652(n)/(2^n)) = A000043(n) - n = A090748(n) - n + 1.

A135656 Perfect numbers divided by 2, written in base 2.

Original entry on oeis.org

11, 1110, 11111000, 111111100000, 111111111111100000000000, 11111111111111111000000000000000, 111111111111111111100000000000000000, 111111111111111111111111111111100000000000000000000000000000

Offset: 1

Omar E. Pol, Feb 28 2008

The number of divisors of a(n) is equal to the number of its digits. This number is equal to 2*A000043(n)-2. The number of divisors of a(n) that are powers of 2 is equal to the number of divisors that are multiples of n-th Mersenne prime A000668(n) and this number of divisors is equal to A090748(n). The first digits of a(n) are "1". For n>1 the last digits are "0". The number of digits "1" is equal to A000043(n). The number of digits "0" is equal to A000043(n)-2. The concatenation of digits "1" gives the n-th Mersenne prime written in binary (see A117293(n)). The structure of divisors of a(n) represent a triangle (see example).

			a(4)=111111100000 because the 4th. perfect number is 8128 and 8128/2=4064 and 4064 written in base 2 is 111111100000. Note that 1111111 is the 4th. Mersenne prime A000668(4)=127, written in base 2.
The structure of divisors of a(4)=111111100000

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Perfect numbers divided by 2: A133028. Cf. A000396, A000668, A019279, A090748, A117293, A135650.

a(n)=A133028(n) written in base 2.

cp's OEIS Frontend

A020492 Balanced numbers: numbers k such that phi(k) (A000010) divides sigma(k) (A000203).

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A134708 Even superperfect numbers divided by 2.

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A293391 Integers n such that sigma(n)/phi(n) is a perfect square.

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A139045 Largest proper divisor of the Fibonacci numbers > 1.

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A134705 a(n) = n-th perfect number divided by 2^n.

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A134709 Even superperfect numbers divided by 2, minus 1.

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A134710 a(n) = n-th even superperfect number divided by 2^n.

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